Goodness of fit test - overview

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The population proportions in each of the $J$ conditions are $\pi_1$, $\pi_2$, $\ldots$, $\pi_J$

or equivalently

The probability of drawing an observation from condition 1 is $\pi_1$, the probability of drawing an observation from condition 2 is $\pi_2$, $\ldots$,
the probability of drawing an observation from condition $J$ is $\pi_J$

Alternative hypothesis

The population proportions are not all as specified under the null hypothesis

or equivalently

The probabilities of drawing an observation from each of the conditions are not all as specified under the null hypothesis

Assumptions

Sample size is large enough for $X^2$ to be approximately chi-squared distributed. Rule of thumb: all $J$ expected cell counts are 5 or more

Sample is a simple random sample from the population. That is, observations are independent of one another

Test statistic

$X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
where the expected cell count for one cell = $N \times \pi_j$, the observed cell count is the observed sample count in that same cell, and the sum is over all $J$ cells

Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$

Example context

Is the proportion of people with a low, moderate, and high social economic status in the population different from $\pi_{low}$ = .2, $\pi_{moderate}$ = .6, and $\pi_{high}$ = .2?

SPSS

Analyze > Nonparametric Tests > Legacy Dialogs > Chi-square...

Put your categorical variable in the box below Test Variable List

Fill in the population proportions / probabilities according to $H_0$ in the box below Expected Values. If $H_0$ states that they are all equal, just pick 'All categories equal' (default)

Jamovi

Frequencies > N Outcomes - $\chi^2$ Goodness of fit

Put your categorical variable in the box below Variable

Click on Expected Proportions and fill in the population proportions / probabilities according to $H_0$ in the boxes below Ratio. If $H_0$ states that they are all equal, you can leave the ratios equal to the default values (1)